Functional Valued Padé-type Approximant And Degeneracies Of Generalised Inverse Funtion-valued Padé Approximant And Applications In Integral Equations

This article consists of three parts.In the first part, theory and method of the function-valued Padé-type approximant (FPTA) is studied. The function-valued Padé-type approximant is defined by using introducing a function-valued linear functional on polynomial space, then it is applied to solve the second kind of Fredholm integral equations. The method of FPTA in this paper has better approximation effect at the poles of the given series than the previous methods. Moreover, due to the degree of its denominator polynomials may be odd or even order number, so FPTA fills up the gap of the generalised inverse function-valued Pade approximant (GIPA). Four efficient algorithms to compute FPTA are established, respectively. Some examples are given to show these algorithms are efficient and useful. Two kinds of error formulas of FPTA are proved. In the end, the row and column convergence problems for FPTA are discussed in detail and two sufficient conditiions are given.In the second part, the degeneracy cases of generalised inverse function-valued pade approximant (GIPA) are at first discussed. The degeneacy cases means that the denominator polynomial of GIPA has odd degree or has zeroes of odd order at the origin or both. At first, the extended GIPA is defined, which enlarges the domain of GIPA. Its existence and uniqueness theorems are proved. Second, GIPA in the degeneracy cases of type [n — σ/2k — 2σ] are constructed. In the end, the characteristic of the table for the extended GIPA is presented. The study about the degeneracy cases greatly enriches the theory of function-valued Padé-type approximation.In the third part, the application problems of GIPA and FPTA are discussed. In section one, two new methods, which is called the real part method of ε-algorithm of GIPA and the determinant method of orthogonal polynomials of FPTA, respectively, are presented to accelerate convergence of the given function sequences. In section two, two new methods estimating the characteristic values of the second kind of Fredholm integral equations are presented by means of taking its real part of E-algorithm and taking the zeros of the determinant of orthogonal polynomials, respectively. Some exapmles are given to illustrate above methods.